1 /* mpfr_expm1 -- Compute exp(x)-1
2
3 Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4 Contributed by the AriC and Caramel projects, INRIA.
5
6 This file is part of the GNU MPFR Library.
7
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 3 of the License, or (at your
11 option) any later version.
12
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
17
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
22
23 #define MPFR_NEED_LONGLONG_H
24 #include "mpfr-impl.h"
25
26 /* The computation of expm1 is done by
27 expm1(x)=exp(x)-1
28 */
29
30 int
mpfr_expm1(mpfr_ptr y,mpfr_srcptr x,mpfr_rnd_t rnd_mode)31 mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
32 {
33 int inexact;
34 mpfr_exp_t ex;
35 MPFR_SAVE_EXPO_DECL (expo);
36
37 MPFR_LOG_FUNC
38 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
39 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
40 inexact));
41
42 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
43 {
44 if (MPFR_IS_NAN (x))
45 {
46 MPFR_SET_NAN (y);
47 MPFR_RET_NAN;
48 }
49 /* check for inf or -inf (expm1(-inf)=-1) */
50 else if (MPFR_IS_INF (x))
51 {
52 if (MPFR_IS_POS (x))
53 {
54 MPFR_SET_INF (y);
55 MPFR_SET_POS (y);
56 MPFR_RET (0);
57 }
58 else
59 return mpfr_set_si (y, -1, rnd_mode);
60 }
61 else
62 {
63 MPFR_ASSERTD (MPFR_IS_ZERO (x));
64 MPFR_SET_ZERO (y); /* expm1(+/- 0) = +/- 0 */
65 MPFR_SET_SAME_SIGN (y, x);
66 MPFR_RET (0);
67 }
68 }
69
70 ex = MPFR_GET_EXP (x);
71 if (ex < 0)
72 {
73 /* For -1 < x < 0, abs(expm1(x)-x) < x^2/2.
74 For 0 < x < 1, abs(expm1(x)-x) < x^2. */
75 if (MPFR_IS_POS (x))
76 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
77 else
78 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {});
79 }
80
81 MPFR_SAVE_EXPO_MARK (expo);
82
83 if (MPFR_IS_NEG (x) && ex > 5) /* x <= -32 */
84 {
85 mpfr_t minus_one, t;
86 mpfr_exp_t err;
87
88 mpfr_init2 (minus_one, 2);
89 mpfr_init2 (t, 64);
90 mpfr_set_si (minus_one, -1, MPFR_RNDN);
91 mpfr_const_log2 (t, MPFR_RNDU); /* round upward since x is negative */
92 mpfr_div (t, x, t, MPFR_RNDU); /* > x / ln(2) */
93 err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ?
94 MPFR_EMIN_MIN : -LONG_MAX) <= 0 ?
95 - (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) :
96 - mpfr_get_si (t, MPFR_RNDU);
97 /* exp(x) = 2^(x/ln(2))
98 <= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon))
99 with epsilon > 0 */
100 mpfr_clear (t);
101 MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, minus_one, err, 0, 0, rnd_mode,
102 expo, { mpfr_clear (minus_one); });
103 mpfr_clear (minus_one);
104 }
105
106 /* General case */
107 {
108 /* Declaration of the intermediary variable */
109 mpfr_t t;
110 /* Declaration of the size variable */
111 mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */
112 mpfr_prec_t Nt; /* working precision */
113 mpfr_exp_t err, exp_te; /* error */
114 MPFR_ZIV_DECL (loop);
115
116 /* compute the precision of intermediary variable */
117 /* the optimal number of bits : see algorithms.tex */
118 Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
119
120 /* if |x| is smaller than 2^(-e), we will loose about e bits in the
121 subtraction exp(x) - 1 */
122 if (ex < 0)
123 Nt += - ex;
124
125 /* initialize auxiliary variable */
126 mpfr_init2 (t, Nt);
127
128 /* First computation of expm1 */
129 MPFR_ZIV_INIT (loop, Nt);
130 for (;;)
131 {
132 MPFR_BLOCK_DECL (flags);
133
134 /* exp(x) may overflow and underflow */
135 MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN));
136 if (MPFR_OVERFLOW (flags))
137 {
138 inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS);
139 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
140 break;
141 }
142 else if (MPFR_UNDERFLOW (flags))
143 {
144 inexact = mpfr_set_si (y, -1, rnd_mode);
145 MPFR_ASSERTD (inexact == 0);
146 inexact = -1;
147 if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1))
148 {
149 inexact = 1;
150 mpfr_nexttozero (y);
151 }
152 break;
153 }
154
155 exp_te = MPFR_GET_EXP (t); /* FIXME: exp(x) may overflow! */
156 mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* exp(x)-1 */
157
158 /* error estimate */
159 /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
160 err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);
161
162 if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
163 {
164 inexact = mpfr_set (y, t, rnd_mode);
165 break;
166 }
167
168 /* increase the precision */
169 MPFR_ZIV_NEXT (loop, Nt);
170 mpfr_set_prec (t, Nt);
171 }
172 MPFR_ZIV_FREE (loop);
173
174 mpfr_clear (t);
175 }
176
177 MPFR_SAVE_EXPO_FREE (expo);
178 return mpfr_check_range (y, inexact, rnd_mode);
179 }
180